On idempotent measure conjecture and decomposition of invariant measures

TL;DR

This paper explores the structure of invariant measures in NIP theories, focusing on the *-product, f-generic types, and the Idempotent Measure Conjecture, providing new insights into measure decomposition and classification.

Contribution

It introduces a novel approach to classify ergodic measures and analyze the Idempotent Measure Conjecture within the framework of NIP theories.

Findings

Classification of ergodic Keisler measures in NIP theories

Analysis of minimal ideals of invariant types

Insights into the Idempotent Measure Conjecture

Abstract

We work with the *-product introduced in [GHK25] and f-generic types to describe the minimal ideals of invariant types and to classify ergodic Keisler measures in amenable NIP theories. Moreover, we analyze the situation around the so-called Idempotent Measure Conjecture studied in [CGK24] and [GHK25].